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We measure the Born cross sections of the process e+e−→K+K−K+K− at center-of-mass (c.m.) energies, s, between 2.100 and 3.080 GeV. The data were collected using the BESIII detector at the BEPCII collider. An enhancement at s=2.232 GeV is observed, very close to the e+e−→ΛΛ¯ production threshold. A similar enhancement at the same c.m. energy is observed in the e+e−→ϕK+K− cross section. The energy dependence of the K+K−K+K− and ϕK+K− cross sections differs significantly from that of e+e−→ϕπ+π−.